Computing Nonsimple Polygons of Minimum Perimeter

Fekete, Sándor P.; Haas, Andreas; Hemmer, Michael; Hoffmann, Michael M.; Kostitsyna, Irina; Krupke, Dominik; Maurer, Florian; Mitchell, Joseph S. B.; Schmidt, Arne; Schmidt, Christiane; Troegel, Julian

doi:10.1007/978-3-319-38851-9_10

Cited by 3 publications

(1 citation statement)

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“…Also, since the number of variables in the new proposed model is exponential in the cardinality of the input point set, we resort to the use of column generation, which leads to the development of a branch-and-price algorithm. Since ILP is often applied in Operations Research, but much less frequently in Computational Geometry, this article can be seen as a further contribution towards bridging these two communities [8,16,10,12,13,26,30].…”

Section: Our Contributionmentioning

confidence: 99%

Solving the Minimum Convex Partition of Point Sets with Integer Programming

Sapucaia¹,

Rezende²,

Souza³

2020

Preprint

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The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map -which represents, say, a geographical area -into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (mcpp). A convex partition of a point set P in the plane is a subdivision of the convex hull of P whose edges are segments with both endpoints in P and such that all internal faces are empty convex polygons. The mcpp is an NP-hard problem where one seeks to find a convex partition with the least number of faces.We present a novel polygon-based integer programming formulation for the mcpp, which leads to better dual bounds than the previously known edge-based model. Moreover, we introduce a primal heuristic, a branching rule and a pricing algorithm. The combination of these techniques leads to the ability to solve instances with twice as many points as previously possible while constrained to identical computational resources. A comprehensive experimental study is presented to show the impact of our design choices.

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Section: Our Contributionmentioning

confidence: 99%

Solving the Minimum Convex Partition of Point Sets with Integer Programming

Sapucaia¹,

Rezende²,

Souza³

2020

Preprint

View full text Add to dashboard Cite

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Computing Nonsimple Polygons of Minimum Perimeter

Fekete

Haas

Hemmer

et al. 2016

Experimental Algorithms

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We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP) asks for a (not necessarily simply connected) polygon with shortest possible boundary length. Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MPP. On the positive side, we show how to achieve a constant-factor approximation. When trying to solve MPP instances to provable optimality by means of integer programming, an additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting additional geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that using a natural geometry-based sparsification yields results that are on average within 0.5% of the optimum.

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